L^p-Integrability of Radon-Nikodym Densities Between Harmonic Energy Measures on the Sierpinski Gasket

2026/07/24 by Konstantinos Tsougkas
#math.CA #math.FA #math.SP

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Abstract

It is known that the energy measures of any two nonconstant harmonic functions on the standard Sierpiński gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence for L^p-integrability of the corresponding Radon--Nikodym densities in the range 1<p<(log 15)/(log 9). For arbitrary ordered pairs of nonconstant harmonic functions, we prove uniform boundedness of the associated density-ratio power sums, and hence L^p-integrability, in the subinterval 1<p<(log(35/3))/(log 9). When the denominator harmonic direction is represented by the boundary values (0,-1,1), we prove boundedness throughout the full conjectured interval.

Citations

• Non-degeneracy of the harmonic structure on Sierpinski Gaskets
• The Kusuoka measure and the energy Laplacian on level-k Sierpiński gaskets
• Energy Measures of Harmonic Functions on the Sierpiński Gasket
• Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals

Related

• Energy Measures of Harmonic Functions on the Sierpiński Gasket
• Cheeger's energy on the Harmonic Sierpinski Gasket
• Density of Lipschitz functions in Energy
• A Fourier restriction theorem based on convolution powers
• Hahn decomposition and Radon-Nikodym theorem with a parameter